Monoidal categories graded by partial commutative monoids
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908893620207616 |
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| author | Earnshaw, Matthew Nester, Chad Román, Mario |
| author_facet | Earnshaw, Matthew Nester, Chad Román, Mario |
| contents | Effectful categories have two classes of morphisms: pure morphisms, which form a monoidal category; and effectful morphisms, which can only be combined monoidally with central morphisms (such as the pure ones), forming a premonoidal category. This suggests seeing morphisms of an effectful category as carrying a grade that combines under the monoidal product in a partially defined manner. We axiomatize this idea with the notion of monoidal category graded by a partial commutative monoid (PCM). Monoidal categories arise as the special case of grading by the singleton PCM, and effectful categories arise from grading by a two-element PCM. Further examples include grading by powerset PCMs, modelling non-interfering parallelism for programs accessing shared resources, and grading by intervals, modelling bounded resource usage. We show that effectful categories form a coreflective subcategory of PCM-graded monoidal categories; introduce cartesian structure, recovering Freyd categories; and describe PCM-graded monoidal categories as monoids by viewing a PCM as a thin promonoidal category. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16375 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Monoidal categories graded by partial commutative monoids Earnshaw, Matthew Nester, Chad Román, Mario Logic in Computer Science Category Theory Effectful categories have two classes of morphisms: pure morphisms, which form a monoidal category; and effectful morphisms, which can only be combined monoidally with central morphisms (such as the pure ones), forming a premonoidal category. This suggests seeing morphisms of an effectful category as carrying a grade that combines under the monoidal product in a partially defined manner. We axiomatize this idea with the notion of monoidal category graded by a partial commutative monoid (PCM). Monoidal categories arise as the special case of grading by the singleton PCM, and effectful categories arise from grading by a two-element PCM. Further examples include grading by powerset PCMs, modelling non-interfering parallelism for programs accessing shared resources, and grading by intervals, modelling bounded resource usage. We show that effectful categories form a coreflective subcategory of PCM-graded monoidal categories; introduce cartesian structure, recovering Freyd categories; and describe PCM-graded monoidal categories as monoids by viewing a PCM as a thin promonoidal category. |
| title | Monoidal categories graded by partial commutative monoids |
| topic | Logic in Computer Science Category Theory |
| url | https://arxiv.org/abs/2603.16375 |